Prove that the difference between two consecutive square numbers is always odd. - OCR - GCSE Maths - Question 18 - 2017 - Paper 1
Question 18
Prove that the difference between two consecutive square numbers is always odd.
Worked Solution & Example Answer:Prove that the difference between two consecutive square numbers is always odd. - OCR - GCSE Maths - Question 18 - 2017 - Paper 1
Step 1
Step 1: Define Consecutive Square Numbers
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Answer
Let the two consecutive integers be represented as ( n ) and ( n + 1 ). The square of the first integer is ( n^2 ) and the square of the second integer is ( (n + 1)^2 ).
Step 2
Step 2: Express the Difference
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Answer
The difference between the two squares can be represented as follows:
[ (n + 1)^2 - n^2 ]
Expanding this gives:
[ (n^2 + 2n + 1) - n^2 = 2n + 1 ]
Step 3
Step 3: Analyze the Result
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Answer
The expression ( 2n + 1 ) is composed of an even number ( 2n ) plus 1, which results in an odd number. Therefore, the difference between two consecutive square numbers is always odd.
Step 4
Step 4: Conclusion
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Answer
Thus, we have proven that the difference between any two consecutive square numbers, ( n^2 ) and ( (n + 1)^2 ), is always odd.
